Discrete mathematics

Physics

Undecidable problem

Pure mathematics

Mathematics

Limit (mathematics)

Decidability

Finite set

Counterexample

Mathematical analysis

Artificial intelligence

Geometry

Computer science

Tensor product

Limit set

Quantum mechanics

Product (mathematics)

Embedding

Hilbert space

Quantum

Forum of Mathematics Pi

We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.

THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED